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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.3.12
7장, 문제 7.3.12

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.


tanh(−x) = −tanh x

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Step 1: Recall the definition of the hyperbolic tangent function: tanh(x)=ex-e-xex+e-x.
Step 2: Substitute -x into the definition of tanh. This gives: tanh(-x)=e-x-exe-x+ex.
Step 3: Factor out -1 from the numerator of the fraction: tanh(-x)=-(ex-e-x)e-x+ex.
Step 4: Recognize that the numerator is the negative of the numerator in the definition of tanh(x). Thus, tanh(-x)=-tanh(x).
Step 5: Conclude that the identity tanh(-x)=-tanh(x) is proven using the definition of hyperbolic functions.

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Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are based on hyperbolas rather than circles. The primary hyperbolic functions include sinh(x), cosh(x), and tanh(x), which are defined as sinh(x) = (e^x - e^(-x))/2, cosh(x) = (e^x + e^(-x))/2, and tanh(x) = sinh(x)/cosh(x). Understanding these definitions is crucial for proving identities involving hyperbolic functions.
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5:50
Asymptotes of Hyperbolas

Odd and Even Functions

An odd function is defined by the property f(-x) = -f(x), while an even function satisfies f(-x) = f(x). The hyperbolic tangent function, tanh(x), is an odd function, which means that tanh(-x) = -tanh(x). This property is essential for proving identities involving tanh, as it allows for simplifications when substituting negative arguments.
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06:21
Properties of Functions

Proof Techniques in Calculus

Proof techniques in calculus often involve direct substitution, algebraic manipulation, and the application of definitions. To prove identities like tanh(−x) = −tanh x, one typically starts with the left-hand side, applies the definition of tanh, and uses properties of exponents and odd functions to arrive at the right-hand side. Mastery of these techniques is vital for successfully demonstrating mathematical identities.
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Fundamental Theorem of Calculus Part 1
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Tsunamis A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging from 150 to 1000 km. Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about 4000 m. Explain why the shallow-water velocity equation (Exercise 75) applies to tsunamis even though the actual depth of the water is large. What does the shallow-water equation say about the speed of a tsunami in the Pacific Ocean (use d = 4000 m)?

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Rule of 70 Bankers use the Rule of 70, which says that if an account increases at a fixed rate of p%/yr, its doubling time is approximately 70/p. Use linear approximation to explain why and when this is true.

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37–56. Integrals Evaluate each integral.

∫ cosh 2x dx

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Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

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22–36. Derivatives Find the derivatives of the following functions.


f(t) = 2 tanh⁻¹ √t

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Uranium dating Uranium-238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and they determine that 85% of the original U-238 remains; the other 15% has decayed into lead. How old is the rock?

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